arXiv 2201.111582022-01math.AP

欧拉流中二维点涡的稳定性

Stability of the two-dimensional point vortices in Euler flows

Dengjun Guo

备注：: 28 pages

摘要：

考虑二维不可压Euler方程\[\begin{cases} \partial_t ω+ u\cdot \nabla ω=0 \\ ω(0,x)=ω_0(x). \end{cases}\]我们感兴趣的是初始涡量具有$ω_0=ω_{0,ε}+ω_{0p,ε}$形式的情况，其中$ω_{0,ε}$集中在$M$不相交点$p_m^0$附近，而$ω_{0p,ε}$是一个小扰动项。首先，我们证明，对于这样的初始涡，解决方案$ω(x,t)$承认分解$ω(x,t)=ω_ε(x,t)+ω_{p,ε}(x,t)$，其中$ω_ε(x,t)$仍然集中在附近的$M$点$p_m(t)$和$ω_{p,ε}(x,t)$仍然很小的$t \in [0,T]$。其次，我们给出了一个定量的描述时，初始涡的形式$ω_0(x)=\sum_{m=1}^M \frac{γ_m}{ε^2}η(\frac{x-p_m^0}ε)$，我们不假设$η$有紧凑的支持。最后，我们证明，如果$p_m(t)$保持分离的所有$t\in[0,+\infty)$，那么$ω(x,t)$仍然集中在附近的$M$点至少为$t \le c_0 |\log A_ε|$，其中$A_ε$是小的，并收敛到$0$作为$ε\to 0$。

英文摘要：

We consider the two-dimensional incompressible Euler equation \[\begin{cases} \partial_t ω+ u\cdot \nabla ω=0 \\ ω(0,x)=ω_0(x). \end{cases}\] We are interested in the cases when the initial vorticity has the form $ω_0=ω_{0,ε}+ω_{0p,ε}$, where $ω_{0,ε}$ is concentrated near $M$ disjoint points $p_m^0$ and $ω_{0p,ε}$ is a small perturbation term. First, we prove that for such initial vorticities, the solution $ω(x,t)$ admits a decomposition $ω(x,t)=ω_ε(x,t)+ω_{p,ε}(x,t)$, where $ω_ε(x,t)$ remains concentrated near $M$ points $p_m(t)$ and $ω_{p,ε}(x,t)$ remains small for $t \in [0,T]$. Second, we give a quantitative description when the initial vorticity has the form $ω_0(x)=\sum_{m=1}^M \frac{γ_m}{ε^2}η(\frac{x-p_m^0}ε)$, where we do not assume $η$ to have compact support. Finally, we prove that if $p_m(t)$ remains separated for all $t\in[0,+\infty)$, then $ω(x,t)$ remains concentrated near $M$ points at least for $t \le c_0 |\log A_ε|$, where $A_ε$ is small and converges to $0$ as $ε\to 0$.

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英文摘要：

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